Descriptor Systems (DAEs) – Linear and Nonlinear Examples

This is an Invited Lecture that I presented at the Online Workshop “Applications of Algebra in Science and Engineering (AASE)”, which was organised by the Dept. of Mathematics, National Institute of Technology, Jamshedpur, India, during October 2020. At the time, I was doing my PostDoc at the Lanscom group of the Physics Department, Aristotle University of Thessaloniki. This video is also available in the Lab's Youtube channel

Here is a list of references with useful examples of Descriptor Systems:
-- Tripathi, M., Moysis, L., Gupta, M. K., Fragulis, G. F., & Volos, C. (2024). Observer Design for Nonlinear Descriptor Systems: A Survey on System Nonlinearities. Circuits, Systems, and Signal Processing, 1-20.

1. Efstathios N Antoniou, Athanasios A Pantelous, Ioannis A Kougioumtzoglou, and Antonina Pirrotta. Response determination of linear dynamical systems with singular matrices: A polynomial matrix theory approach. Applied Mathematical Mo delling, 42:423–440, 2017.
2. José M Araújo, Péricles R Barros, and Carlos ET Dorea. Design of observers with error limitation in discrete-time descriptor systems: A case study of a hydraulic tank system. IEEE Transactions on Control Systems Technology, 20(4):1041–1047, 2012.
3. A. A Belov, O. G Andrianova, and Alexander P Kurdyukov. Control of Discrete-Time Desc riptor Systems: An Anisotropy-Based Approach. Springer, 2018.
4. Angelika Bunse-Gerstner, Ralph Byers, Volker Mehrmann, and Nancy K Nichols. Feedback design for regularizing descriptor systems. Linear algebra a nd its applications, 299(1-3):119–151, 1999.
5. S. L Campbell and Peter Kunkel. General nonlinear differential algebraic equations and tracking problems: A robotics example. 2018.
6. C. Coll, A. Herrero, E. Sánchez, and N.Thome. A dynamic model for a study of diabetes. Mathematical and Computer Modelling, 50(5-6):713–716, 2009.
7. C.Coll, A. Herrero, E. Sánchez, and N. Thome. An optimal control for a diabetes model. MODELLING FOR MEDICINE, BUSINESS AND ENGINEERING 2009, page 50, 2009.
8. M. Gupta, N. Tomar, V. Kumar Mishra, and S. Bhaumik. Observer design for semilinear descriptor systems with applications to chaos-based secure communication. International Journal of Applie d and Computational Mathematics, 3(1):1313–1324, 2017.
9. C. Liu, Q. Zhang, Y. Zhang, and X. Duan. Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator. International Journal of Bifurc ation and Chaos, 18(10):3159–3168, 2008.
10. P. Liu, Q. Zhang, X. Yang, and L. Yang. Passivity and optimal control of descriptor biological complex systems. IEEE Trans. Autom. Control, 53:122–125, 2008.
11. L. Moysis, A. Pantelous, E. Antoniou, and N. Karampetakis. Closed form solution for the equations of motion for constrained linear mechanical systems and generalizations: An algebraic approach. J Franklin Inst, 354(3):1421–1445, 2016.
12. F. Pasqualetti, F. Dörfler, and F. Bullo. Attack detection and identification in cyber-physical systems. IEEE Transactions on Automatic Control, 58(11):2715–2729, 2013.
13. T. Reis. Mathematical modeling and analysis of nonlinear time-invariant rlc circuits. In Large-Scale Networks in Engineering and Life Sciences, pages 125–198. Springer, 2014.
14. R. Riaza. Daes in circuit modelling: a survey. In Surveys in Differential-Algebraic Equations I. Springer, 2013.
15. R. Riaza and C. Tischendorf. Semistate models of electrical circuits including memristors. International Journal of Circuit Theory and Applications, 39(6):607–627, 2011.
16. F. Udwadia and P. Phohomsiri. Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics.
Proceedings: Mathematical, Physical and Engineering Sciences, pages 2097–2117, 2006.
17. G. Verghese, B. Lévy, and T. Kailath. A generalized state-space for singular systems. IEEE Transactions on Automatic Control, 26(4):811–831, 1981.
18. H. Wang, Z. Han, W. Zhang, and Q. Xie. Chaotic synchronization and secure communication based on descriptor observer. Nonlinear Dynamics, 57(1-2):69, 2009.
19. H.Wang, X.Zhu, S. Gao, and Z. Chen. Singular observer approach for chaotic synchronization and private communication. Communications in Nonlinear Science and Numerical Simulation, 16:1517–1523, 2011.
20. G. Zhang, Y. Shen, and B. Chen. Bifurcation analysis in a discrete differential-algebraic predator–prey system. Applied Mathematical Modelling, 38(19-20):4835–4848, 2014.
21. G. Zhang, L. Zhu, and B. Chen. Hopf bifurcation and stability for a differential-algebraic biological economic system. Applied Mathematics and Computation, 217(1):330–338, 2010.
22. Moysis, L., Tripathi, M., Gupta, M., & Volos, C. (2021,). Chaos Synchronization, Anti-Synchronization, and Parameter Estimation in a Chaotic System with Coexisting Hidden Attractors. (ICC). IEEE.

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